A more complicated approach is to prove the result by induction. Let
$$S(n)=\frac{1}{2^2}+\frac{1}{3^2}+\cdots +\frac{1}{n^2}.$$
We want to show that $S(n)\lt 1-\dfrac{1}{n}$ for every integer $n \ge 2$.

The result is clearly true when $n=2$. We show that for any $k\ge 2$, if the result is true when $n=k$, it is true when $n=k+1$.
We have
$$S(k+1)=S(k)+\frac{1}{(k+1)^2}.$$
By the induction assumption, $S(k)\lt 1-\dfrac{1}{k}$. It follows that
$$S(k+1)\lt 1-\frac{1}{k}+\frac{1}{(k+1)^2}.$$
But
$$\frac{1}{k}-\frac{1}{(k+1)^2}=\frac{k^2+k+1}{k(k+1)^2}\gt \frac{k^2+k}{k(k+1)^2}=\frac{1}{k+1}.$$
It follows that $S(k+1)\lt 1-\dfrac{1}{k+1}$.

Remark: If we try to prove the weaker result $S(n)\lt 1$ by induction, we get into trouble. For from the induction assumption $S(k)\lt 1$, we cannot conclude in any direct way that $S(k+1)\lt 1$. So with the above problem, we have the seemingly paradoxical fact that in an induction proof, a strong inequality can be easier to prove than a weaker inequality.